Fast and fault-tolerant logical measurements: Auxiliary hypergraphs and transversal surgery

TL;DR

This work develops a theory of fast, fault-tolerant quantum code surgery using auxiliary hypergraphs. It introduces block reading as a concrete, transversal, multi-block surgery achieving amortised constant-time logical measurements while preserving code distance, and extends to partial block reading and general hypergraph surgery with intermediate time overhead via locally testable codes and modular expansion. A circuit- and ZX-calculus–based equivalence between surgery and homomorphic measurement clarifies how auxiliary connectivity governs fault tolerance, culminating in general bounds on measurement time and precision. Collectively, these results reveal that reducing measurement time hinges on the connectivity between a code and its measurement ancilla rather than single-shot memory capabilities, enabling parallel, scalable logical measurements for CSS LDPC codes with controlled space-time overhead.

Abstract

Quantum code surgery is a promising technique to perform fault-tolerant computation on quantum low-density parity-check codes. Recent developments have significantly reduced the space overhead of surgery. However, generic surgery operations still require $O(d)$ rounds of repeated syndrome extraction to be made fault-tolerant. In this work, we focus on reducing the time overhead of surgery. We first present a general set of conditions that ensure fault-tolerant surgery operations can be performed with constant time overhead. This fast surgery necessarily makes use of an auxiliary complex described by a hypergraph rather than a graph. We then introduce a concrete scheme called block reading, which performs transversal surgery across multiple code blocks. We further investigate surgery operations with intermediate time overhead, between $O(1)$ and $O(d)$, which apply to quantum locally testable codes. Finally, we establish a circuit equivalence between homomorphic measurement and hypergraph surgery and derive bounds on the time overhead of generic logical measurement schemes. Overall, our results demonstrate that reducing the time cost of code surgery is not reliant on the quantum memory being single-shot. Instead it is chiefly the connectivity between a code and its measurement ancilla system that determines the achievable measurement time overhead.

Figures (9)

• Figure 1: Examples of scalable Tanner graphs.
• Figure 2: Constructing a thickened hypergraph. Note that (b) is a conventional Tanner graph, not a scalable Tanner graph.
• Figure 3: Example of partial block reading with a thickened hypergraph. The logical measurement is the same as in Eq. \ref{['eq:three_blocks_partial']}, but the hypergraph has been thickened to prevent the weight of $\overline{Z}$ operators dropping below $d$.
• Figure 4: The same partial block reading as in Figure \ref{['fig:thickened_partial_reading']} but with additional redundant checks and metachecks.
• Figure 5: A fault complex for a unit cell of the 2D surface code, starting and terminating at $X$-type boundaries. Time flows from bottom to top.

Theorems & Definitions (99)

• Theorem 1.1: Fast hypergraph surgery, informal statement of Theorem \ref{['thm:genhypersurg']}
• Theorem 1.2: Full block reading, informal statement of Theorem \ref{['thm:full_block_amortised']}
• Theorem 1.3: Partial block reading, informal statement of Theorem \ref{['thm:fault_distance_many_subcode_1']} and Corollary \ref{['coro:partial_BR_reduced_space_soundness']}
• Theorem 1.4: Intermediate time hypergraph surgery, informal statement of Theorem \ref{['thm:fault_distance_hypergraph_together']}
• Theorem 1.5: Informal statement of results in Section \ref{['sec:hom_equivalence']}
• Theorem 1.5
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4